Which of the following numbers when added to 7, will be divisible by 11?

Options

A. 115
B. 56
C. 3
D. 2006

Explanation

To solve the problem of which number, when added to 7, will be divisible by 11, we can follow these steps:

### Step 1: Understand the Problem
We need to find a number \( x \) such that when we add it to 7, the result \( 7 + x \) is divisible by 11. This can be expressed mathematically as:
\[ 7 + x \equiv 0 \ (\text{mod} \ 11) \]

### Step 2: Rearranging the Equation
From the equation above, we can rearrange it to find \( x \):
\[ x \equiv -7 \ (\text{mod} \ 11) \]

### Step 3: Finding the Equivalent Positive Value
To find a positive equivalent of \(-7\) in modulo 11, we can add 11 to \(-7\):
\[ -7 + 11 = 4 \]
Thus, we have:
\[ x \equiv 4 \ (\text{mod} \ 11) \]

### Step 4: Check the Options
Now, we will check which of the provided options, when added to 7, gives a result that is equivalent to 4 modulo 11.

1. **Option 1**: \( x = 3 \)
\[ 7 + 3 = 10 \quad (10 \mod 11 = 10) \]
Not divisible by 11.

2. **Option 2**: \( x = 10 \)
\[ 7 + 10 = 17 \quad (17 \mod 11 = 6) \]
Not divisible by 11.

3. **Option 3**: \( x = 56 \)
\[ 7 + 56 = 63 \quad (63 \mod 11 = 8) \]
Not divisible by 11.

4. **Option 4**: \( x = 115 \)
\[ 7 + 115 = 122 \quad (122 \mod 11 = 1) \]
Not divisible by 11.

5. **Option 5**: \( x = 4 \)
\[ 7 + 4 = 11 \quad (11 \mod 11 = 0) \]
This is divisible by 11.

### Conclusion
The number that, when added to 7, results in a sum that is divisible by 11 is **4**.

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